Hasimoto surfaces for two classes of curve evolution in Minkowski 3-space

Abstract In this work, we study Hasimoto surfaces for the second and third classes of curve evolution corresponding to a Frenet frame in Minkowski 3-space. Later, we derive two formulas for the differentials of the second and third Hasimoto-like transformations associated with the repulsive-type nonlinear Schrödinger equation.


Introduction
Many mathematicians have been interested for a long time in studying connections between integrable equations (soliton equations) and geometric motions of a curve in various spaces. In particular, Hasimoto [1] discovered a connection between thin vortex filament without stretching in an incompressible inviscid fluid and the nonlinear Schrödinger equation. Lakshmanan [2] presented a connection between the nonlinear Schrödinger equation and integrable Landau-Lifshitz equation for a time evolution of a spin vector X as follows: Recently, some geometers studied connections between moving frame of space curves and soliton equations in ambient spaces (see [4][5][6][7][8][9][10][11][12][13]).
On the other hand, Lamb [3] introduced the Hasimoto transformation as a complex function and he studied a certain class of moving space curves with soliton equations. Murugesh and Balakrishnan [5] presented two Hasimoto transformations as two other complex functions. Also, they showed that there are two other classes of curve evolution that may be so identified. Hence, three distinct classes of curve evolution are associated with the nonlinear Schrödinger equation in Euclidean 3-space. The differential formulas of Hasimoto transformations in Euclidean 3-space have been presented by Langer and Perline [4]. For the extension to the Lorentz version, Gürbüz suggested three classes of curve evolution associated with the nonlinear Schrödinger equation in Minkowski 3-space (see [6,8,9,12]). Also, Gürbüz [6] extended the results of Langer and Perline in [4] in Minkowski 3-space, and she derived differential formulas of Hasimoto transformations for the first class of curve evolutions associated with the repulsive-type nonlinear Schrödinger equation in Minkowski 3-space.
In this paper, we give differential formulas of two Hasimoto-like transformations of the second and third classes associated with the repulsive-type nonlinear Schrödinger equation in Minkowski 3-space. 2 Hasimoto surface for second class

Time evolution of second class
Let β be a spacelike curve parametrized by arc-length s in Minkowski 3-space 1 3 . Then, the Frenet frame , , satisfies the following formulas: Now we take the second Hasimoto-like transformation Ψ and the second Darboux vector ξ 2 as follows, respectively: Considering the derivatives of { } P P P , , ⁎ with respect to s and u, and using the second Hasimoto-like transformation, we obtain, respectively: We also have Here f g , 2 2 , and h 2 are smooth functions. Then, In other words, the vortex filament flow for the second class associated with − NLS is defined as Using (2.4) and (2.6), we have Thus, from (2.2), (2.3), and (2.7), we obtain the following results.

Second Hasimoto-like transformation
Consider a vector field along a spacelike curve with a timelike binormal vector for the second class of curve evolution associated with − NLS . Since Z 2 satisfies the arc length preserving condition, On the other hand, a second normalization operator N 2 for the second class according to a Frenet frame is introduced as: Also, the second linear recursion operator R 2 for the second class is given by: (2.11) The differential formula of the second Hasimoto-like transformation ( ) = H β Ψ 2 can be presented as follows: Then from (2.11) and (2.13), the second linear recursion operator is derived as: (2.14) From (2.12) and (2.14), we have the following theorem.    In other words, the vortex filament flow for the third class associated with the − NLS equation is defined as: (3.9) Thus, we obtain the following theorem.

Conclusion
In this paper, Hasimoto surfaces for the second and third classes of a spacelike curve evolution were studied. Also, two formulas for the differentials of two Hasimoto-like transformations corresponding to a Frenet frame in Minkowski 3-space have been derived.